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On the n-Coupling Problem

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  • Rüschendorf, Ludger
  • Uckelmann, Ludger

Abstract

In this paper we obtain based on an idea of M. Knott and C. S. Smith (1994, Linear Algebra Appl.199, 363-371) characterizations of solutions of three-coupling problems by reduction to the construction of optimal couplings of each of the variables to the sum. In the case of normal distributions this leads to a complete solution. Under a technical condition this idea also works for general distributions and one obtains explicit results. We extend these results to the n-coupling problem and derive a characterization of optimal n-couplings by several 2-coupling problems. This leads to some constructive existence results for Monge solutions.

Suggested Citation

  • Rüschendorf, Ludger & Uckelmann, Ludger, 2002. "On the n-Coupling Problem," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 242-258, May.
    • Handle: RePEc:eee:jmvana:v:81:y:2002:i:2:p:242-258
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    References listed on IDEAS

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    1. Rüschendorf, L. & Rachev, S. T., 1990. "A characterization of random variables with minimum L2-distance," Journal of Multivariate Analysis, Elsevier, vol. 32(1), pages 48-54, January.
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    Cited by:

    1. Bernard, Carole & Chen, Jinghui & Rüschendorf, Ludger & Vanduffel, Steven, 2023. "Coskewness under dependence uncertainty," Statistics & Probability Letters, Elsevier, vol. 199(C).
    2. Henri Heinich, 2006. "The Monge Problem in Banach Spaces," Journal of Theoretical Probability, Springer, vol. 19(2), pages 509-534, June.
    3. Valentina Masarotto & Victor M. Panaretos & Yoav Zemel, 2019. "Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 172-213, February.
    4. Ludger Rüschendorf, 2012. "Worst case portfolio vectors and diversification effects," Finance and Stochastics, Springer, vol. 16(1), pages 155-175, January.
    5. Henry-Labordère, Pierre & Tan, Xiaolu & Touzi, Nizar, 2016. "An explicit martingale version of the one-dimensional Brenier’s Theorem with full marginals constraint," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2800-2834.
    6. Puccetti, Giovanni & Rüschendorf, Ludger & Vanduffel, Steven, 2020. "On the computation of Wasserstein barycenters," Journal of Multivariate Analysis, Elsevier, vol. 176(C).

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